Parameterized Inapproximability for Steiner Orientation by Gap Amplification

In the k-Steiner Orientation problem, we are given a mixed graph, that is, with both directed and undirected edges, and a set of k terminal pairs. The goal is to find an orientation of the undirected edges that maximizes the number of terminal pairs for which there is a path from the source to the sink. The problem is known to be W[1]-hard when parameterized by k and hard to approximate up to some constant for FPT algorithms assuming Gap-ETH. On the other hand, no approximation factor better than ?(k) is known. We show that k-Steiner Orientation is unlikely to admit an approximation algorithm with any constant factor, even within FPT running time. To obtain this result, we construct a self-reduction via a hashing-based gap amplification technique, which turns out useful even outside of the FPT paradigm. Precisely, we rule out any approximation factor of the form (log k)^o(1) for FPT algorithms (assuming FPT ? W[1]) and (log n)^o(1) for purely polynomial-time algorithms (assuming that the class W[1] does not admit randomized FPT algorithms). This constitutes a novel inapproximability result for polynomial-time algorithms obtained via tools from the FPT theory. Moreover, we prove k-Steiner Orientation to belong to W[1], which entails W[1]-completeness of (log k)^o(1)-approximation for k-Steiner Orientation. This provides an example of a natural approximation task that is complete in a parameterized complexity class. Finally, we apply our technique to the maximization version of directed multicut - Max (k,p)-Directed Multicut - where we are given a directed graph, k terminals pairs, and a budget p. The goal is to maximize the number of separated terminal pairs by removing p edges. We present a simple proof that the problem admits no FPT approximation with factor ?(k^(1/2 - ?)) (assuming FPT ? W[1]) and no polynomial-time approximation with ratio ?(|E(G)|^(1/2 - ?)) (assuming NP ? co-RP)

A Survey on Approximation in Parameterized Complexity: Hardness and Algorithms

Parameterization and approximation are two popular ways of coping with NP-hard problems. More recently, the two have also been combined to derive many interesting results. We survey developments in the area both from the algorithmic and hardness perspectives, with emphasis on new techniques and potential future research directions

Gap Amplification for Reconfiguration Problems

In this paper, we demonstrate gap amplification for reconfiguration problems. In particular, we prove an explicit factor of PSPACE-hardness of approximation for three popular reconfiguration problems only assuming the Reconfiguration Inapproximability Hypothesis (RIH) due to Ohsaka (STACS 2023). Our main result is that under RIH, Maxmin Binary CSP Reconfiguration is PSPACE-hard to approximate within a factor of $0.9942$. Moreover, the same result holds even if the constraint graph is restricted to $(d,\lambda)$-expander for arbitrarily small $\frac{\lambda}{d}$. The crux of its proof is an alteration of the gap amplification technique due to Dinur (J. ACM, 2007), which amplifies the $1$ vs. $1-\epsilon$ gap for arbitrarily small $\epsilon > 0$ up to the $1$ vs. $1-0.0058$ gap. As an application of the main result, we demonstrate that Minmax Set Cover Reconfiguration and Minmax Dominating Set Reconfiguratio} are PSPACE-hard to approximate within a factor of $1.0029$ under RIH. Our proof is based on a gap-preserving reduction from Label Cover to Set Cover due to Lund and Yannakakis (J. ACM, 1994). However, unlike Lund--Yannakakis' reduction, the expander mixing lemma is essential to use. We highlight that all results hold unconditionally as long as "PSPACE-hard" is replaced by "NP-hard," and are the first explicit inapproximability results for reconfiguration problems without resorting to the parallel repetition theorem. We finally complement the main result by showing that it is NP-hard to approximate Maxmin Binary CSP Reconfiguration within a factor better than $\frac{3}{4}$.Comment: 41 pages, to appear in Proc. 35th Annu. ACM-SIAM Symp. Discrete Algorithms (SODA), 202

LIPIcs, Volume 248, ISAAC 2022, Complete Volume

LIPIcs, Volume 248, ISAAC 2022, Complete Volum

FPT-approximation for FPT problems

Over the past decade, many results have focused on the design of parameterized approximation algorithms for W[1]-hard problems. However, there are fundamental problems within the class FPT for which the best known algorithms have seen no progress over the course of the decade. In this paper, we expand the study of FPT-approximation and initiate a systematic study of FPT-approximation for problems that are in FPT. We design FPT-approximation algorithms for problems that are in FPT, with running times that are significantly faster than the corresponding best known FPT-algorithm, and while achieving approximation ratios that are significantly better than what is possible in polynomial time

LIPIcs, Volume 244, ESA 2022, Complete Volume

LIPIcs, Volume 244, ESA 2022, Complete Volum